Abstract
The reliability matrix, being an oblique projection operator, transforms correlated observations into the least squares residuals in Gauss–Markov models. It also allows to study model responses in individual observations to the assumed configurations of gross errors. The variability of the basic characteristics of the operator due to the increase in observation correlations is investigated by means of numerical tests and theoretical derivations. The characteristics such as diagonal elements and asymmetry indices have not that long ago been introduced as the response-based measures of internal reliability and subjected to the analysis. Here, additionally, the relationship between the asymmetry indices and the angles of non-orthogonality of projection is derived. The measures are compared in terms of the effect of observation correlations with the commonly used reliability measures obtained on the basis of statistical tests for detection and identification of outliers, such as generalized reliability numbers and minimal detectable biases. For the purposes of the present paper, the latter are termed the testing-based measures. The comparative analysis shows that both the types, when taken together, provide complete information about the behaviour of a GMM with correlated observations in the presence of a gross error in a particular observation and about its detectability. Hence, the conclusion is that the response-based measures can be a useful supplementation of the testing-based measures for the phase of network design.
Highlights
The covariance matrices for observations are a basis for constructing the stochastic models for satellite and ground positioning systems
The covariances have an effect upon the elements of reliability matrices. It was already found in Wang and Chen (1994) and Schaffrin (1997) that at some level of observation correlations there may appear on a main diagonal of a reliability matrix the negative elements as well as the elements greater than 1
For better determining of the effect itself, several networks with different levels of internal reliability will be used in the tests
Summary
The covariance matrices for observations are a basis for constructing the stochastic models for satellite and ground positioning systems. The covariances have an effect upon the elements of reliability matrices. It was already found in Wang and Chen (1994) and Schaffrin (1997) that at some level of observation correlations there may appear on a main diagonal of a reliability matrix the negative elements as well as the elements greater than 1. It was noticed that the diagonal elements being one-dimensional quantities do not fully describe the responses of a model to gross errors in the observations. The proposal of a generalized reliability number appeared. Further study of this problem led to the so-called response-based reliability measures
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